potential function

Given an electrical network $G = (V, E)$ , a potential function $U : V \to R$ is an assignment such that for all $u \overset{e}{\to} v$ , $V_{e} = U_{v} - U_{e}$ , defined up to scaling by a constant.
The existence of such a function is equivalent to Kirchoff's second law

The nice thing about this formulation is that it's independent of direction.

Ohm's law then may be rewritten as

I_{e} = \frac{U_{v} - U_{u}}{R_{e}}

and the first law is equivalent to for all vertices $v_{i}$ connected to $v$ for $v$ not connected to the battery

\sum_{i} \frac{U_{v} - U_{v_{i}}}{R_{i}} = 0

We can combine all the $U_{v}$ terms together to get

(\frac{1}{R_{1}} + \dots + \frac{1}{R_{d}}) U_{v} - \sum_{i} \frac{U_{v_{i}}}{R_{i}} = 0

If $v = 1$ , then this would be equal to $I_{b a t}$
If $v = n$ , then this would be equal to $- I_{b a t}$

Connection to Kirchoff's matrix

The first law is equivalent to saying that for $\vec{U}$ the vector of all potentials $\vec{U} = [U_{1}, \dots, U_{n}]^{T}$

K \vec{U} = [\begin{matrix} I_{b a t} \\ 0 \\ ⋮ \\ 0 \\ - I_{b a t} \end{matrix}] $ $ w h e r e $ K $ i s [[Z e t t e l k a s t e n / U n f i l e d / K i r c h o f f^{'} s m a t r i x o f a n e l e c t r i c a l n e t w o r k ∥ K i r c h o f f^{'} s m a t r i x o f a n e l e c t r i c a l n e t w o r k]] . H o w e v e r, $ K $ i s d e g e n e r a t e a s t h e r e a r e m a n y s o l u t i o n s f o r $ \vec{U} $ u p t o s c a l i n g . T h u s, w e " g r o u n d " t h e n e t w o r k b y s e t t i n g $ U_{n} = 0 $ .

\tilde{K} \begin{bmatrix}
U_1 \
\vdots \
U_{n-1}
\end{bmatrix} = \begin{bmatrix}
I_{bat} \
0 \
\vdots \
0
\end

w h e r e $ \tilde{K} $ i s t h e [[Z e t t e l k a s t e n / U n f i l e d / r e d u c e d L a p l a c i a n m a t r i x ∥ r e d u c e d L a p l a c i a n m a t r i x]] (a s w e r e m o v e d t h e $ n $ t h r o w a n d c o l u m n) . F u r t h e r m o r e, w e c a n m a k e t h e a d d i t i o n a l s i m p l i f y i n g a s s u m p t i o n t h a t $ I_{b a t} = 1 $, t h e n r e s c a l e a f t e r w a r d s . T h e n, n o t e t h a t t h e [[Z e t t e l k a s t e n / U n f i l e d / e f f e c t i v e r e s i s t a n c e ∥ e f f e c t i v e r e s i s t a n c e]] i s t h e n g i v e n b y $ U_{1} $, o f w h i c h w e m a y s o l v e w i t h [[Z e t t e l k a s t e n / U n f i l e d / C r a m e r^{'} s r u l e ∥ C r a m e r^{'} s r u l e]] .